Spectral networks are a combinatorial tool consisting of labelled lines on a Riemann surface. They have a surprising amount of applications and are intimately linked to non-Abelianization of flat connections, Fock–Goncharov cluster coordinates, exact WKB theory, etc. After reviewing this story for the $SL(2)$ and $SL(3)$ case, I will describe this is in detail for the group $G_2$. Time permitting, I will give as an application a concrete parametrization of the nonabelian Hodge correspondence for the Hitchin component of the split real form of $G_2$. This is joint work with Andy Neitzke.

I will describe a map that associates to every deformation of an object in a higher category a collection of generalized symmetries of the object. Building on work by Lurie, we will see that the failure of this map to be an equivalence can be quantified. Under favorable circumstances, the map is an equivalence, and this leads to an explicit description of the space of deformations in terms of solutions to certain equations. I will discuss applications of these results to topological field theory and holomorphic symplectic geometry. This talk is based on joint work with Bhanu Kiran.

In recent years, there has been a growing number of applications of stable homotopy theory to condensed matter physics, many of which stem from a conjecture of Kitaev that gapped invertible phases of matter should be classified by the homotopy groups of a spectrum. This gives rise to a mathematical modeling question: how do we model quantum systems in such a way that this result can be better understood, perhaps even proved? In this talk, I will discuss some aspects of this modeling problem. This is based on joint work with Mike Hermele, Juan Moreno, Markus Pflaum, Marvin Qi and Daniel Spiegel, David Stephen, Xueda Wen.